# Parametric Question

• Jan 21st 2010, 05:19 PM
xwrathbringerx
Parametric Question
Hi

P(2ap, ap^2), Q(2aq,aq^2) and R(2ar,ar^2) are points on the parabola x^2 = 4ay.
a) Show that the equation of the normal at P is py+x = 2ap + ap^3.
b) Find the coordinates of the point of intersection of the normals at P and Q.

c) If the normals P,Q and R are constant, show that p +q +r=0.

How on earth do you do this??

• Jan 22nd 2010, 07:08 AM
Quote:

Originally Posted by xwrathbringerx
Hi

P(2ap, ap^2), Q(2aq,aq^2) and R(2ar,ar^2) are points on the parabola x^2 = 4ay.
a) Show that the equation of the normal at P is py+x = 2ap + ap^3.
b) Find the coordinates of the point of intersection of the normals at P and Q.

c) If the normals P,Q and R are constant, show that p +q +r=0.

How on earth do you do this??

hi

(a) Use the chain rule to get the gradient of tangent , then use m1m2=-1 for the gradient of normal

$\displaystyle x=2ap\Rightarrow \frac{dx}{dp}=2a$

$\displaystyle y=ap^2\Rightarrow \frac{dy}{dp}=2ap$

so $\displaystyle \frac{dy}{dx}=\frac{dy}{dp}\cdot \frac{dp}{dx}=2ap\cdot \frac{1}{2a}=p$

Equation : $\displaystyle y-ap^2=-\frac{1}{p}(x-2ap)$
(b) Since Q is also a point on the parabola , its equation of normal would be $\displaystyle qy+x=aq^3+2aq$ , just swap the 'p' with 'q' , then have them equal to solve for the point of intersection .